Department of Computer Science
Faculty of Mathematics, Physics and Informatics
Comenius University in Bratislava
Abstract model of the effectivity of human problem solving
with respect to cognitive psychology and computational complexity
Autoreferát dizerta£nej práce
Franti²ek „uri²
na získanie vedecko-akademickej hodnosti philosophiae doctor
v odbore doktorandského ²túdia : 9.2.1 Informatika
Bratislava, 2015
Dizerta£ná práca bola vypracovaná v dennej forme doktorandského ²túdia na Katedre informatiky Fakulty matematiky, fyziky a informatiky Univerzity Komenského v Bratislave.
Predkladate©:
Franti²ek „uri²
Katedra informatiky FMFI UK
Mlynská dolina, 842 48 Bratislava
’kolite©:
prof. RNDr. Pavol „uri², CSc.
Katedra informatiky FMFI UK
Mlynská dolina, 842 48 Bratislava
Oponenti:
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Obhajoba dizerta£nej práce sa koná d¬a ................... o ................... hod. pred komisiou pre
obhajobu dizerta£nej práce v odbore doktorandského ²túdia vymenovanou predsedom odborovej
komisie d¬a ................... v ²tudijnom odbore
9.2.1 Informatika
na Fakulte matematiky, fyziky a informatiky Univerzity Komenského v Bratislave.
Predseda odborovej komisie:
prof. RNDr. Branislav Rovan, PhD.
Katedra informatiky FMFI UK
Mlynská dolina, 842 48 Bratislava
1 Motivation and goals of the thesis
Problem solving is probably the most common activity of all organisms, especially of humans.
We deal with various problems throughout our lives, from infancy to adulthood. Therefore, it
is not surprising that an extensive eort has been made to understand the cognitive processes
responsible for this ability.
One possible research approach is to look at why some people are able to solve particular
classes of problems while others are not, how they go about it, what makes some problems
insolvable, and how is it possible that a small supercial change can make a very dicult problem
into a trivial one (and vice versa). Through this approach psychologists and other scientists were
able to shed a lot of light on many cognitive mechanisms of human problem solving.
However, there is dierence between solving a problem by discovering a crucial relation that
allows a quick solution and an extensive and long search through many possibilities. Undoubtedly,
human beings have immense capacity for solving vast number of diverse and complex problems,
but the pinnacle of our mental potential is not the capacity itself but rather the eectivity of
this capacity. In this sense, we believe that there is still little research done on the eectivity of
human problems solving, that is, we need to have a closer look at why some people are able to
solve problems (much) faster than others, how and why are they able to discern and focus on
the information crucial to the solution, and generally what this eectivity depends on and how
good it really is when compared with some provable eective method.
In this thesis we attempt to give answers to the last two questions. And while these answers,
if proved correct, provide new and important insights into human problem solving, they are
of high value to AI research as well. Human brain is so far our only example of an eective
general problem solving system (Langley, 2006), and by understanding the principles behind its
eectivity, the development in AI (like cognitive architectures) can advance along more specic
paths towards attaining human level abilities.
Here we formulate more particularly two our goals (mentioned above), and add some related
ones:
1. Identify the cognitive processes (mechanisms, abilities) sucient for successful human
problem solving, and extract those of them that are the sources/roots of the eectivity of
human problem solving
2. Compare the eectivity of human problem solving with an optimal strategy for problem
solving (Solomono, 1986)
3. Using the previous results, propose an algorithm for problem solving
4. Analyse the optimal Solomono strategy when it is used mistakenly by imprecise information (note that it is used mistakenly quite often, since to gain enough precise information
may be a hard problem)
The goals 1, 2, 3, and 4 have been achieved by the results from Sections 3 & 4, from Section
5, from Section 6, and from Section 7, respectively).
3
2 Introduction
Generally, the process of problem solving can be simplied to nding, testing, and applying
the ideas (solution candidates). The eective problem solving can be then viewed as applying
the right ideas at the right time. Such ideas can be consciously sought for, or they can come
unconsciously like an automated procedure (tying shoelaces, for example). One way to nd these
ideas, albeit usually very ineective one, is the aforementioned Blind search. On one hand, this
exhaustive search is sometimes inevitable. For example, any problem with encrypted assignment
is for a solver without the decryption key (or extensive cryptanalytic skills) practically insolvable
without the exhaustive search (in fact, its insolvable either way as the space of decryption keys
is usually huge, but that's not the point). Similarly, solving Rubik cube by trying all possible
move sequences is highly inecient as 20 moves are sucient to solve any instance.
Therefore, Blind search is a universal but not an eective method, if there are too many
solution candidates to check them one by one (which is usually the case). The other source of
its ineectivity is that the candidates are not tested in any problem related way (hence the
name Blind search). Solomono (1986) in his problem solving system exploited a theorem in
probability stating that if an appropriate order could be imposed on the solution candidates,
then checking them in this order would yield probabilistically optimal solution strategy.
(Solomono, 1986). Let m1 , m2 , m3 , ... be candidates/ideas (or, in mechanical
problem solving, strings) that can be used to solve a problem. Let pk be probability that the
candidate mk will solve the problem, and tk the time required to generate and test this candidate.
Then, testing the candidates in the decreasing order ptkk gives the minimal expected time before a
solution is found.
Theorem 2.1
Corollary 2.2.
The eectivity of problem solving depends on
1. Knowledge and experience,
2. Ability to generate − in the eective order (Theorem 2.1) and in a
appropriate candidate ideas for solving the (sub)problems.
short time − the
3 Cognitive mechanisms related to the eective human problem
solving
We propose a list of six cognitive abilities or mechanisms that, we argue, signicantly help
humans to generate in the eective order and in short time the appropriate candidate ideas
for solving the (sub)problems. Thus, in our opinion these are the mechanisms that give rise
to the eectivity of human problem solving, and as such should be implemented in cognitive
architectures for this reason.
The following processes (mechanisms, abilities) are related to the eectivity of
human problem solving (with respect to Theorem 2.1, especially to the second point of Corollary
2.2):
Proposition 3.1.
4
1.
Discovering similarities
2.
Discovering relations, connections and associations
3.
Generalization, specication
4.
Abstraction
5.
Intuition
6.
Context sensitivity and the ability to focus and forget
4 General human problem solving mechanisms
In this section we analysed general human problem solving process and identied necessary and
probably sucient mechanisms for successful human problem solving.
The following processes (mechanisms, abilities) are necessary for successful
human problem solving
Proposition 4.1.
1.
Discovering similar concepts (representing information, experience, properties, problems, situations, models, ...)
2.
Discovering related, connected, and associated concepts (representing information,
experience, properties, problems, situations, models, ...)
3.
Manipulation1 with the information, problem, or situation or its representation
(e.g., transform, split, add or remove concepts, features, attributes, imagine, experiment,
...)
4.
Assessing1 the situation/progress and deciding what to do next
5.
Incubation (stop solving the problem)
Furthermore, other cognitive processes not directly linked with problem solving, including
(a) language processing,
(b) working memory processes (coordinating, monitoring, and executing the intended activities),
(c) ability to interpret/understand memories and experience,
are (most likely) used as well.
The processes (mechanisms, abilities) from the Proposition 4.1 are sucient
for successful human problem solving.
Hypothesis 4.2.
1
By this we mean application of some method, procedure, heuristic, experience, common sense, logic, ... that
performs manipulation/assessment action on something.
5
Additionally, we linked the eective cognitive mechanisms from Proposition 3.1 with general
problem solving mechanisms from Proposition 4.1, thus establishing rst arguments about the
eectivity of human problem solving.
In terms of human problem solving capabilities, the processes (mechanisms,
abilities) from Proposition 3.1 together with (a)(c) from Proposition 4.1 suce to replace the
processes (mechanisms, abilities) from Proposition 4.1.
Proposition 4.3.
Tthe processes (mechanisms, abilities) from Proposition 3.1 together with the
cognitive processes for
Hypothesis 4.4.
(a) language processing,
(b) working memory process (coordinating, monitoring, and executing the intended activities),
(c) ability to interpret/understand memories and experience,
are sucient for successful human problem solving. Together, they are the mechanisms for general
human problem solving.
5 The eectivity of human problem solving
In this section we put together our results from the Sections 3 and 4 to analyse the scope and
the roots of the eectivity of human problem solving.
Hypothesis 5.1.
In the probabilistic sense of Theorem 2.1, human problem solving is eective
with respect to
1. solver's knowledge and experience,
2. quality of his processes, mechanisms, or abilities from Proposition 3.1.
Given what we observe in the world, the human problem solving process is indeed fast (i.e.,
eective). Whence this eectivity comes from is still uncertain, but the results of our work
summarized in Hypothesis 5.1 suggests that the roots of the eectivity of human problem
solving lie in
1.
2.
3.
4.
optimal problem solving strategy from Solomono (1986),
solver's knowledge and experience,
quality of the solver's mechanisms from Proposition 3.1,
language processing (in the sense of Baldo et al., 2005).
6 Human problem solving model
In this section we describe the eective general human problem solving process as an algorithm
based on the general problem solving mechanisms from the Section 4 and eective cognitive
mechanisms from Section 3.
6
Proposition 6.1.
The human problem solving can be formulated as a process of the following
steps.
1.
Represent the problem and identify the diculty
2.
Asses the problem model for appropriateness and eectivity through macro process 4
from Proposition 4.1, or, if formulated as a separate problem (e.g., "How do I assess my
problem model?"), through all mechanisms from Proposition 4.12 .
(a) If a sucient model is available, go to step 3.
(b) If there are more available sucient models, select a subset and go to step 3.
(c) Otherwise, formulate a new problem of nding better model (or of transforming the
problem to a more promising problem3 ) , and go to step 1 (new problem to be solved).
3.
Find and asses idea(s) to continue solving the problem (within the current model 4 )
(a) If an applicable idea is available through the mechanisms from Proposition 4.1, go to
step 4.
(b) If more applicable ideas are available, select a subset and go to step 4.
(c) Otherwise, formulate a new problem (of nding better idea, model, or of transforming
the problem to a more promising problem), and go to step 1 (new problem to be solved).
apply the (selected) idea5 , if necessary interact with the world, and update
the problem model accordingly.
4. Parallely
5. If the problem is not solved,
return to step 2 or 3, or (temporarily) give up.
7 Mathematical aspects of the eective problem solving
In this section we consider the eect of interchanging two candidates with respect to the optimal
Solomono strategy (Theorem 2.1) on the problem solving time and the number of candidates
examined. We give several bounds on the error resulting from the mentioned interchange. However, since the values pi and ti from Theorem 2.1 can be arbitrary, we examine three special
restrictions (called expert, novice, and indierent system, respectively) under which reasonable
bounds can be achieved. Finally, we consider a modication of the Solomono strategy when
the value of ti for each i is not xed. This modication models the case when we applied the
same solution candidate (e.g., a method) to two or more similar problems each time solving the
problem in dierent times.
2
If new problem is formulated, the algorithm recursively iterates.
That is, an easier, more known, simplied version of the problem, or its decomposition into sub-problems
(e.g., independent divide & conquer, dependent dynamic programming)
4
If there are more models selected, follow them by rotation.
5
If there are more problem solving ideas selected, follow them by rotation.
3
7
(Solomono, 1986). If each bet costs 1 dollar, then betting in the order of decreasing value pk (i.e., always taking the bet with highest win probability available) would give the
greatest win probability per dollar.
Theorem 7.1
Note that the expected number of solution candidates examined is not given by
ES because we did not include the possibility that all of our solution candidates failed to solve
the problem. The corrected value ES is given by
Remark 7.2.
ES =
N
X
i·
i=1
i−1
Y
(1 − pj ) · pi + N
j=1
N
Y
(1 − pj ).
j=1
This is because the probability of each candidate failing to solve the problem is
while it takes us N trials to discover this.
QN
j=1 (1
− pj ),
(Solomono, 1986). In the general scenario, if one continues to select subsequent
bets on the basis of maximum pk /dk , the expected money spent before winning will be minimal.
Suppose we change dollars to some measure of time (tk ). Then, betting according to this strategy
yields the minimum expected time to win.
Theorem 7.3
Note that the expected problem solving time is not given by ET because, again,
we did not include the possibility that all of our solution candidates failed to solve the problem.
The corrected value ET is given by
Remark 7.4.
ET =
N X
i
X
i=1 l=1
tl ·
i−1
Y
(1 − pj ) · pi +
j=1
N
X
l=1
tl ·
N
Y
(1 − pj )
j=1
for the same reasons as in Remark 7.2.
Let pk − pk+1 = θ > 0 for some k ∈ {1, 2, ..., N − 1} (assuming {pi }N
i=1 to
be ordered as before in the proof of the Theorem 7.1). Then, following the optimal Solomono
strategy from Theorem 7.1 with (k +1)th solution candidate tried just before k th (a solver's error)
yields a sub-optimal expected number of solution candidates tried before either nding a solution
or discovering that none of our solution candidates works, and the expected excess EXC can be
quantied as follows
k−1
Y
EXC =
(1 − pj ) · θ.
Theorem 7.5.
j=1
Furthermore,
k−1
2k−4
).
θ · e−Sk−1 ≥ EXC ≥ θ · (1 − Sk−1 + (k − 2)Pk−1
Exchanging the k th and (k + n)th solution candidates in the optimal Solomono
strategy from Theorem 7.1 (a solver's error) increases the expected number of solution candidates
examined by at most the excess
EXC = v1 + v2 + v3
Theorem 7.6.
8
where
v1 = k · (pk+n − pk ) · Qk−1 ,
v2 =
k+n−1
pk − pk+n X
·
l · Ql−1 · pl ,
1 − pk
l=k+1
pk − pk+n
· Qk+n−1 .
1 − pk
v3 = (k + n) ·
Let pk − pk+n = θ > 0. Then, the term EXC from Theorem 7.6 can be upper
bounded as follows
θ
EXC ≤
· (k + n) · (npk+1 + 1) e−Sk .
1 − pk
Corollary 7.7.
Theorem 7.8.
Let
pk
pk+1
−
=θ>0
tk
tk+1
for some k ∈ {1, 2, ..., N − 1} (assuming { ptii }N
i=1 to be ordered as before in Theorem 7.3). Then
following the optimal Solomono strategy from Theorem 7.3 with (k + 1)th solution candidate
tried just before k th (a solver's error) yields a sub-optimal expected amount of time spent before
either nding a solution or discovering that none of our solution candidates works, and the
expected excess EXC can be quantied as follows
EXC =
k−1
Y
(1 − pj ) · tk tk+1 · θ.
j=1
Furthermore,
θ · tk tk+1 · e
−Sk−1
k−1
2k−4
≥ EXC ≥ θ · tk tk+1 · 1 − Sk−1 + (k − 2)Pk−1
.
Exchanging the k th and (k + n)th solution candidates in the optimal Solomono
strategy from Theorem 7.3 (a solver's error) increases the expected amount of time by at most
the excess
EXC = q1 + q2 + q3
Theorem 7.9.
where
q1 = Tk−1 · Qk−1 · (pk+n − pk ) + Qk−1 · (tk+n pk+n − tk pk ),
q2 =
k+n−1
X
l=k+1
1 − pk+n
pk − pk+n
Ql−1 · pl Tl ·
+ (tk+n − tk )
,
1 − pk
1 − pk
q3 = Tk+n · Qk+n−1 ·
pk − pk+n
.
1 − pk
9
The rst special case we would like to examine are domain experts (being a domain expert
denitely helps problem solving). We can model the domain expert solver as a system of solution
candidates where each solution candidate has at least some chance of solving a problem. That
is,
∀k : pk ≥ c, for some c ∈ (0, 1).
We are interested in upper bounds on the excess term EXC from Theorems 7.6 and 7.9.
The expected number of excessive solution candidates tried in the expert system
before either nding a solution or discovering that none of our solution candidates works, which
is expressed by the term EXC in Theorem 7.6, can be upper bounded as follows
Theorem 7.10.
EXC ≤ θ
pk+1 (1 − c)k
A − B(1 − c)n−1
2
1 − pk
c
where θ = pk − pk+n , and
"
1 − pk c
A = 1 + kc 1 −
1 − c pk+1
B = (1 − c) + c(n + k) 1 −
1 − p1
1−c
c
k−1 #
,
pk+1
.
Let T be the constant specied above. Then, expected increase of time in the
expert system before either nding a solution or discovering that none of our solution candidates
works, which is expressed by the term EXC in Theorem 7.9, can be upper bounded as follows.
Theorem 7.11.
If pk+n − pk ≤ 0, then
EXC ≤ T · pmax2 ·
pk − pk+n (1 − c)k
A − B(1 − c)n−1
2
1 − pk
c
where
"
1 − pk c
A = 1 + kc 1 −
1 − c pmax2
B = (1 − c) + c(n + k) 1 −
1 − pmax1
1−c
c
pmax2
k−1 #
,
,
pmax1 = max{p1 , ..., pk−1 },
pmax2 = max{pk+1 , ..., pk+n−1 }.
If pk+n − pk ≥ 0, then
EXC ≤ T ·
pk+n − pk
(1 − c)k A − B(1 − pmax )n−1
1 − pk
10
where
A=k
B=
1 − pk
− (1 + kpmax )
1−c
(1 − pmax )k+n−1
(1 − c)k
1 − pmax
1−c
k+n−
c
p2max
k
c
,
p2max
(1 + pmax (k + n − 1)) ,
pmax = max{p1 , ..., pk+n−1 }.
Similarly, we can model the domain novice solver as a system where each solution candidate
has at most some chance succeeding. That is,
∀k : pk ≤ d, for some d ∈ (0, 1).
In this case we are interested in lower bounds on the excess term EXC from Theorems 7.6 and
7.9.
The expected number of excessive solution candidates tried in the novice system
before either nding a solution or discovering that none of our solution candidates works, which
is expressed by the term EXC from the Theorem 7.6, can be lower bounded as follows
Theorem 7.12.
EXC ≥ θ ·
pk+n−1 (1 − d)k
A − B(1 − d)n−1
2
1 − pk
d
where θ = pk − pk+n , and
"
1 − pk
d
A = 1 + kd 1 −
1 − d pk+n−1
B = (1 − d) + d(n + k) 1 −
1 − pk−1
1−d
k−1 #
,
d
pk+n−1
.
Let T be the constant mentioned above. The expected increase of time before
solving the problem in the novice system, which is expressed by the term EXC from the Theorem
7.9, can be lower bounded as follows.
Theorem 7.13.
If pk+n − pk ≤ 0, then
EXC ≥ T · pmin2 ·
pk − pk+n (1 − d)k
A − B(1 − d)n−1
2
1 − pk
d
where
"
1 − pk d
A = 1 + kd 1 −
1 − d pmin2
B = (1 − d) + d(n + k) 1 −
pmin1 = min{p1 , ..., pk−1 },
pmin2 = min{pk+1 , ..., pk+n−1 }.
11
1 − pmin1
1−d
d
pmin2
,
k−1 #
,
If pk+n − pk ≥ 0, then
EXC ≥ T
pk+n − pk
(1 − d)k A − B(1 − pmin )n−1
1 − pk
where
1 − pk
A=k
− (1 + kpmin ) ·
1−d
(1 − pmin )k+n−1
B=
(1 − d)k
1 − pmin
1−d
k+n−
d
p2min
k
d
,
p2min
(1 + pmin (k + n − 1)) ,
pmin = min{p1 , ..., pk+n−1 }.
We can also consider the case where the probabilities pi are all approximately the same
(denote this value p), and the values ti are arbitrary. This case describes the situation where the
solver has many similarly successful candidates (e.g., lots of very general methods of uncertain
success), and he is required to choose. What eect on the expected problem solving time has
the exchanging two candidates in this case?
Let p be the value mentioned above. The expected increase of time before solving
the problem in the indierent system, which is expressed by the term EXC from the Theorem
7.9, can be approximated as follows
Theorem 7.14.
EXC ≈ (tk+n − tk )(1 − p)k−1 1 + (1 − p) − (1 − p)n+1 .
In real life problem solving a particular solution candidate (e.g., a method) could have
been used to solve multiple similar problems each time consuming a dierent amount of time.
Therefore, when the solver is considering a potential solution candidate, it has one cumulative
probability of success (e.g., based on the experience and the strength of similarity/relatedness
with the current problem model), but it can have multiple application times because of this
possible application to the similar problems in the past. What order of examination of the
solution candidates in this setting leads to the minimal expected time to nd a solution? What
if the solver remembers only an approximate average time?
Let sk be a solution candidate which we in the past applied nk times, and let
tk,j be the execution time of the j th application. Denote the mean execution time of the solution
candidate sk with Etk :
tk,1 + ... + tk,nk
.
Etk =
nk
Theorem 7.15.
If one continues to select subsequent candidates on the basis of maximum pk /Etk , then the
expected time before solving the problem will be minimal (provided the problem can be solved by
one of our candidates).
12
Abstract
The ability to solve problems eectively is one of the hallmarks of human cognition. Yet, in
our opinion it gets far less research focus than it rightly deserves. In this paper we outline a
framework in which this eectivity can be studied; we identify the possible roots and scope of
this eectivity and the cognitive processes directly involved. More particularly, we have observed
that people can use cognitive mechanisms to drive problem solving by the same manner on which
an optimal problem solving strategy suggested by Solomono (1986) is based. Furthermore, we
provide evidence for cognitive substrate hypothesis (Cassimatis, 2006) which states that human
level AI in all domains can be achieved by a relatively small set of cognitive mechanisms. The
results presented in this paper can serve both cognitive psychology in better understanding
of human problem solving processes, and articial intelligence in designing more human like
intelligent agents.
Keywords:
human problem solving, eectivity, mechanisms, articial intelligence, cognitive
architecture
Abstrakt
Jedna z najdôleºitej²ích vlastností ©udského myslenia je ur£ite schopnos´ rie²i´ problémy efektívne. V tejto práci popisujeme súvislosti, v ktorých sa dá táto efektivity skúma´. Identikujeme
potenciálne korene a rozsah tejto efektivity a tieº kognitivne procesy, ktoré sú za ¬u zodpovedné.
Presnej²ie, zistili sme, ºe ©udia vedia pouºíva´ isté kognitívne mechanizmy spôsobom ve©mi podobným optimálnej pravdepodobnostnej strategií na rie²enie problémov, ktorú vo svojej práci
pouºil Solomono (1986). Okrem toho, v práci ponúkame argumenty pre "Cognitive substrate
hypothesis"(Cassimatis, 2006), ktorá hovorí, ºe umelá inteligencia na úrovni £loveka moºe by´
dosiahnutá pomocou relatívne malého po£tu kognitívnych mechanizmov. Výsledky prezentované v tejto práci moºu slúºi´ kognitívnej psychologii pri lep²om porozumení ©udského procesu
rie²enia problémov, ako aj umelej inteligencií pri navrhovaní vysoko inteligentných agentov.
©udské rie²enie problémov, efektivita, mechanizmy, umelá inteligencia, kognitívne architektúry
K©ú£ové slová:
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Published work related to the thesis issues
F. Duris. Error bounds on the probabilistically optimal problem solving strategy.
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